New results of fuzzy implications satisfying I(x, I(y, z)) = I(I(x, y), I(x, z))

Cruz et al. (2018) [10] investigated the fuzzy generalization of Frege's Law: x -> (y -> z) (x -> y) -> (x -> z), i.e., I(x, I(y, z)) = I(I(x, y), I(x, z)), which is called generalized Frege's Law. They showed conditions such that the generalized Frege's Law holds for (S, N)-implications (R-, QL-, D-, (T, N)-, H-, respectively). In this paper, firstly, a new necessary condition such that the generalized Frege's Law holds is given: N-I, the natural negation of I, is not continuous or has no fixed point. Based on this result, some propositions in [10] with contradictory assumptions are pointed out, and a correction is given. Secondly, new solutions of the equation I (x, I(y, z)) = I (I(x, y), I(x, z)) in (S, N)-implications are given. Finally, the necessary and sufficient conditions under which the generalized Frege's Law holds for the (U, N)-implications (f-,g-, T-Power based implications, respectively) are studied. (C) 2020 Elsevier Inc. All rights reserved.